Probabilistic recursive functions

Author:
Irwin Mann

Journal:
Trans. Amer. Math. Soc. **177** (1973), 447-467

MSC:
Primary 60B99; Secondary 02F27, 65C05

MathSciNet review:
0322920

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Abstract: The underlying question considered in this paper is whether or not the purposeful introduction of random elements, effectively governed by a probability distribution, into a calculation may lead to constructions of number-theoretic functions that are not available by deterministic means. A methodology for treating this question is developed, using an effective mapping of the space of infinite sequences over a finite alphabet into itself. The distribution characterizing the random elements, under the mapping, induces a new distribution. The property of a distribution being recursive is defined. The fundamental theorem states that recursive distributions induce only recursive distributions. A function calculated by any probabilistic means is called -calculable. For a class of such calculations, these functions are recursive. Relative to Church's thesis, this leads to an extension of that thesis: Every -effectively calculable function is recursive. In further development, a partial order on distributions is defined through the concept of ``inducing.'' It is seen that a recursive atom-free distribution induces any recursive distribution. Also, there exist distributions that induce, but are not induced by, any recursive distribution. Some open questions are mentioned.

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0322920-7

Article copyright:
© Copyright 1973
American Mathematical Society